1. Technical Field of the Invention
The present invention relates to a CT scanner which examines an object using x-rays in the form of a fan beam while rotating around the object and, more particularly, to a CT scanner which is able to produce high-quality tomograms and rapidly perform arithmetic operations needed to reconstruct images.
2. Description of the Prior Art
FIG. 8 shows the structure of a conventional CT scanner of this kind and a method of measurement. An x-ray source S emits x-rays 2 in the form of a fan beam which covers the region 1 of an object 4 to be examined. A multielement x-ray detector 3 is disposed opposite to the source S. The source S rotates around the object 4 to be examined, together with the detector 3. During this rotation, pulsive x-rays are radiated to the object 4 and pass through it. The transmitted fan beam is detected at an angular interval in the source and the detector rotation by detecting elements 3' that are regularly spaced apart from each other in the detector 3 whenever the source and the detector rotate through a discrete step angle. The data about the irradiated object is collected. Assume neighboring elements 3' of the detector 3 are spaced an angle of P.sub..alpha. from each other and that the x-ray source and the detector rotate a discrete step angle of P.sub..beta. for measurement. The obtained data is given by H (.alpha., .beta.), where .beta. is the angular position of the x-ray source and .alpha. indicates the angular position of each detecting element within the array. The data can be obtained by integrating the x-ray absorption coefficient distribution f (x, y) of the object 4 along the detected x-ray fan beam l (x, y). That is, EQU H (.alpha.,.beta.)=.intg.f (x, y)dl (1)
Filtered back projection is the well-known method of reconstructing a tomogram f (x, y) from the data H (.alpha., .beta.) obtained in this way. For example, see "The Fourier Reconstruction of a Head Section", 1974, IEEE TRans., NS-21 pp. 21-43. Referring to FIG. 9, this filtered back projection comprises the steps of acquiring data (A) by projection of x-rays onto an object to be examined, variously prepocessing the data (B) for correcting the physical characteristics of the detector, then subjecting the data to filtering (C) for correcting blur owing to back projection, and subsequently performing arithmetic operations (D) for back projection to reconstruct a tomogram.
In acquiring the data about the transmitted x-ray through the object as described above, the relation between the corrdinate space used for the arithmetic reconstruction operation and the geometry for detection is represented as shown in FIG. 10 using a cartesian coordinate space. In this figure, the position of an x-ray source S is given by angle .beta.. The position of a beam passing through a picture element E (x, y) within a fan beam radiated from the position S is given by the angle .alpha. that the beam makes with the straight line passing through the center of rotation O from the position S. Let L be the distance between x-ray source S and the picture element E (x, y). The radius of the circule drawn by the source S is indicated by D.
As described above, the data obtained from various directions is subjected to filtering (C) (see FIG. 9) and then the data is arithmetically processed (D) so that the data is directly back-projected in a fan-shaped manner onto a two-dimensional memory space constructed as a cartesian coordinate space. This method is known as direct back projection. According to this method, as described in, for example, U.S. Pat. No. 4,149,247, the data H (.alpha., .beta.) obtained by projected fan beams first undergoes a filtering as given by EQU G (.alpha., .beta.)=.intg.J (.alpha.')W(.alpha.')H (.alpha.-.alpha.', .beta.)d.sub..alpha. ' (2)
where J (.alpha.') is a term for correcting for the nonuniform intervals between detecting elements and is approximately given by EQU J (.alpha.')=cos (.alpha.') (3)
and W is a filter function for removing blur. Then these filtered data are back-projected later. More specifically, referring still to FIG. 10, the position of the x-ray source S and the coordinate E (x, y) that is arithmetically processed for reconstruction are given. The angle .alpha. that the x-ray beam passing through the point E makes is calculated according to the following equations (4) and (5). ##EQU1## Then, weight L is calculated according to the following equation. EQU L.sup.2 =(x-D cos .beta.).sup.2 +(y-D sin .beta.).sup.2 ( 6)
Using G and L.sup.2, data about all the coordinates are summed up from the initial position at which .beta.=0.degree. to the final position at which .beta.=360.degree. according to equation (7) to reconstruct a tomogram. ##EQU2## where 1/L.sup.2 is a weight for correcting a partial fan-beam effect that the x-ray beam directed from the x-ray source S toward the detector 3 experiences.
The two sets of data H (.alpha., .beta.) derived by measurement are quantized at intervals of P.sub..alpha. and P.sub..beta., respectively. Using integers j and m,.beta. is rewritten as EQU .beta.=.beta..sub.0 +P.sub..beta. .times.j=.beta.(j) (j=0, 1, 2, . . . , m-1) (8) EQU m=2.pi./P.sub..beta. ( 9)
where .beta..sub.0 is the initial position of the measurement and j indicates the number given to a projected fan beam. Since j is discretized, equation (7) is modified as ##EQU3## With respect to .alpha., the values of a .alpha. found based on the coordinate (x, y) according to equations (4) and (5) do not always coincide with the measuring points. Therefore, instead of using equation (10), it is the common practice to calculate values according to equation (11), making use of linear interpolation taken at four close points. ##EQU4## where gn (.delta.) is an interpolation function, and i and .delta. are given by EQU i=[.alpha./p.alpha.] (12) EQU .delta.=.alpha.-P.alpha.X i (13)
where [ ] is the Gausian symbol.
By performing these prcessings about every coordinate, the arithmetic operations for back projection are completed. For this purpose, the calculations according to equations (4), (5), (6), and (11) must be repeated many times equal to the number of all the picture elements multiplied by the number of projected beams, i.e., the number of incremental steps. Accordingly, the direct back projection needs an exorbitant amount of calculation. The amount increases further as intervals between the detecting elements are narrowed and the number of picture elements of a tomogram is increased to improve the quality of image. This hinders rapid calculation for reconstructing the image.
U.S. Pat. No. Re. 30947 disclosed another method, known as the re-ordering and re-bining method, in which data items obtained by projection of x-ray fan beams are rearranged to create data equivalent to data obtained by using parallel beams. The resultant data are subjected to filtering (C), and arithmetic operations (D) for back projection are performed (see FIG. 9). Referring back to FIG. 8, the multielement x-ray detector 3 has elements which are regularly and circumferentially spaced P from each other about the x-ray source S. Data H (.alpha., .beta.) is obtained by the detector 3 from the projected fan beams. The re-ordering and re-bining method is now described in detail by referring to FIG. 11, where a two-dimensional coordinate space (see FIG. 12) having two parameters is set. One of the parameters is the distance t between the center of rotation O of both the detector 3 and the x-ray source S and each x-ray beam, the other parameter being the angle .theta. that each beam makes. The angle .alpha. and .beta. and the axis t and .theta. of the two-dimensional coordinate space are interrelated by EQU t=D.times.sin .alpha. (14) EQU .theta.=.alpha.+.beta. (15)
Therefore, the angular intervals P.alpha. between the detecting elements in the direction of the t axis of the two-dimensional coordinate space are reduced as the distance t increases. The data items about the projected object are arranged into S-shaped form on the two-dimensional coordinate space according to the angular position .beta. of the x-ray source, as shown in FIG. 12. The data P (t, .theta.) on the two-dimensional coordinate space are derived from projected parallel beams. After the arithmetic operations, the data P (t, .theta.) are arithmetically processed for back projection so that they correspond to parallel beams, whereby a tomogram is reconstructed. In the case of FIG. 12. the ratio K of the angular intervals P.alpha. between the detecting elements to the incremental steps P.beta. for measurement is equal to 1/2.
However, the re-ordering and re-bining method has a disadvantage in that the data drawn on the two-dimensional coordinate space is given by a curve as shown in FIG. 12. Further, since the detecting positions of the detecting elements are not regularly spaced apart, this conversion to the two-dimensional coordinate space is unable to make the lattice points and the measuring points agree with all data items. In general, one-dimensional interpolation is effected along the direction of .beta., and then other one-dimensional interpolation is carried out to correct inhomogeneity in the direction of .alpha.. Or, these two interpolations are combined to perform two-dimensional interpolation in the directions of .alpha. and .beta.. Thus, the reordering and re-bining method suffers from a improvement in the spatial resolution of the reconstructed tomogram because of the two one-dimensional interpolations or the two-dimensional interpolation. Further, the tomogram contains coarser noise components, thus deteriorating the quality of image.